Simulation results
\(J\) regions, \(L_j\) voxels in region \(j\), \(M\) time points.
\[ X_{jlm} = \mu_j + \eta_{jm} + \gamma_{j\ell m} + \epsilon_{j\ell m} \]
For a region \(j\),
\[ X_{j\ell m} = \mu_j + \eta_{jm} + \gamma_{j\ell m} + \epsilon_{j\ell m} \]
\[\begin{align} \label{eq:SpaTempKern} (C_j)_{ll'} &= K(\big\lVert{v_{j\ell} - v_{j\ell'}}\big\rVert; \nu_j, \phi_j) \nonumber \\ (B_j)_{mm'} &= k_{\gamma_j}H(|m - m'|; \tau_{\gamma_j}) + \sigma^2_{\gamma_j}\delta_{mm'} \nonumber \\ (A)_{mm'} &= H(|m - m'|; \tau_\eta) + \sigma^2_\eta \delta_{mm'} \\ \end{align}\]
\[\begin{align} \quad R &= \begin{pmatrix} 1 & \rho_{12} & \cdots & \rho_{1J} \\ \rho_{12} & 1 & \ddots & \cdots \\ \vdots & \ddots & \ddots & \ddots \\ \rho_{J1} & \vdots & \ddots & 1 \end{pmatrix}, \Omega = SRS, \quad S = \operatorname{diag}(k_{\eta_1}^{1/2}, \dotsc, k_{\eta_J}^{1/2}) \\ \label{eq:varMats} \operatorname{Var}(\epsilon) &= \Sigma_\epsilon := \operatorname{diag}(\sigma^2_{\epsilon_1} I_{ML_1},\dotsc , \sigma^2_{\epsilon_J} I_{ML_J}) \\ \operatorname{Var}(\gamma) &= \Sigma_\epsilon^{1/2}\Lambda \Sigma_\epsilon^{1/2}, \quad \Lambda = \operatorname{diag}(C_1 \otimes B_1,\dotsc, C_J \otimes B_J) \\ \operatorname{Var}(\eta) &= (\tilde{\Sigma}_\epsilon^{1/2}\Omega \tilde{\Sigma}_\epsilon^{1/2})\otimes A \end{align}\]
Stage 1 parameters: \((k_{\gamma_j}, \sigma^2_{\gamma_j}, \tau_{\gamma_j}, \phi_j, \sigma^2_{\epsilon_j})\)
Stage 2 parameters: \((\rho_{jj'}, k_{\eta_j}, k_{\eta_{j'}}, \tau_\eta, \sigma^2_{\eta})\)
Need to choose suitable parameters for our simulations.
We consider various simulation settings to evaluate the performance of our method.
For each simulation setting, we generate data from three regions using real voxel coordinates and \(60\) time points.
The three regions contain \(41\), \(25\), and \(77\) voxels.
Correct specification
| delta | psi | mean | median | mad | rmse | sd |
|---|---|---|---|---|---|---|
| 0.1 | 0.2 | 0.056 | 0.076 | 0.399 | 0.476 | 0.476 |
| 0.1 | 0.5 | 0.030 | 0.039 | 0.407 | 0.494 | 0.491 |
| 0.1 | 0.8 | -0.009 | -0.015 | 0.394 | 0.481 | 0.471 |
| 0.5 | 0.2 | 0.106 | 0.119 | 0.143 | 0.186 | 0.187 |
| 0.5 | 0.5 | 0.141 | 0.169 | 0.203 | 0.238 | 0.235 |
| 0.5 | 0.8 | 0.099 | 0.128 | 0.212 | 0.278 | 0.279 |
| 0.7 | 0.2 | 0.106 | 0.108 | 0.138 | 0.168 | 0.169 |
| 0.7 | 0.5 | 0.107 | 0.150 | 0.155 | 0.183 | 0.184 |
| 0.7 | 0.8 | 0.110 | 0.113 | 0.182 | 0.224 | 0.225 |
| delta | psi | mean | median | mad | rmse | sd |
|---|---|---|---|---|---|---|
| 0.1 | 0.2 | 0.297 | 0.382 | 0.405 | 0.489 | 0.488 |
| 0.1 | 0.5 | 0.244 | 0.262 | 0.367 | 0.452 | 0.442 |
| 0.1 | 0.8 | 0.166 | 0.175 | 0.439 | 0.540 | 0.510 |
| 0.5 | 0.2 | 0.347 | 0.338 | 0.149 | 0.182 | 0.183 |
| 0.5 | 0.5 | 0.364 | 0.355 | 0.147 | 0.184 | 0.184 |
| 0.5 | 0.8 | 0.354 | 0.359 | 0.195 | 0.244 | 0.246 |
| 0.7 | 0.2 | 0.339 | 0.345 | 0.122 | 0.156 | 0.157 |
| 0.7 | 0.5 | 0.345 | 0.366 | 0.120 | 0.152 | 0.152 |
| 0.7 | 0.8 | 0.345 | 0.358 | 0.153 | 0.192 | 0.193 |
| delta | psi | mean | median | mad | rmse | sd |
|---|---|---|---|---|---|---|
| 0.1 | 0.2 | 0.480 | 0.555 | 0.314 | 0.424 | 0.409 |
| 0.1 | 0.5 | 0.369 | 0.449 | 0.343 | 0.456 | 0.395 |
| 0.1 | 0.8 | 0.255 | 0.242 | 0.469 | 0.561 | 0.445 |
| 0.5 | 0.2 | 0.592 | 0.607 | 0.108 | 0.138 | 0.138 |
| 0.5 | 0.5 | 0.582 | 0.591 | 0.150 | 0.184 | 0.184 |
| 0.5 | 0.8 | 0.562 | 0.570 | 0.174 | 0.217 | 0.215 |
| 0.7 | 0.2 | 0.597 | 0.603 | 0.072 | 0.091 | 0.091 |
| 0.7 | 0.5 | 0.586 | 0.601 | 0.110 | 0.142 | 0.142 |
| 0.7 | 0.8 | 0.573 | 0.581 | 0.120 | 0.153 | 0.152 |
The average of intra-regional correlations is given by \[ \alpha_j = \frac{1}{L_j^2} \sum_{\ell, \ell'} \operatorname{Cor}(Y_j(v_{j\ell, t_m}), Y_j(v_{j\ell'}, t_m)) \] We can write \[ \alpha_j = \delta_j + (1-\delta_j)\psi_j \] where
In simulated data, we choose parameters such that \(\delta_j = 0.1, 0.5, 0.7\) and \(\psi_j = 0.2, 0.5, 0.8\).
We consider both temporal and spatial misspecification.
Temporal misspecification
| delta | psi | mean | median | mad | rmse | sd |
|---|---|---|---|---|---|---|
| 0.1 | 0.2 | 0.075 | 0.072 | 0.338 | 0.428 | 0.429 |
| 0.1 | 0.5 | 0.004 | -0.014 | 0.439 | 0.532 | 0.526 |
| 0.1 | 0.8 | -0.075 | -0.138 | 0.498 | 0.585 | 0.561 |
| 0.5 | 0.2 | 0.111 | 0.096 | 0.146 | 0.218 | 0.219 |
| 0.5 | 0.5 | 0.074 | 0.088 | 0.131 | 0.166 | 0.165 |
| 0.5 | 0.8 | 0.063 | 0.056 | 0.192 | 0.249 | 0.247 |
| 0.7 | 0.2 | 0.161 | 0.103 | 0.203 | 0.332 | 0.328 |
| 0.7 | 0.5 | 0.108 | 0.095 | 0.145 | 0.231 | 0.232 |
| 0.7 | 0.8 | 0.108 | 0.097 | 0.179 | 0.277 | 0.278 |
| delta | psi | mean | median | mad | rmse | sd |
|---|---|---|---|---|---|---|
| 0.1 | 0.2 | 0.363 | 0.384 | 0.278 | 0.364 | 0.366 |
| 0.1 | 0.5 | 0.282 | 0.355 | 0.404 | 0.533 | 0.531 |
| 0.1 | 0.8 | 0.189 | 0.242 | 0.469 | 0.579 | 0.559 |
| 0.5 | 0.2 | 0.338 | 0.358 | 0.120 | 0.151 | 0.152 |
| 0.5 | 0.5 | 0.346 | 0.341 | 0.147 | 0.193 | 0.194 |
| 0.5 | 0.8 | 0.374 | 0.363 | 0.175 | 0.242 | 0.242 |
| 0.7 | 0.2 | 0.451 | 0.381 | 0.205 | 0.290 | 0.273 |
| 0.7 | 0.5 | 0.411 | 0.372 | 0.185 | 0.259 | 0.253 |
| 0.7 | 0.8 | 0.380 | 0.359 | 0.157 | 0.219 | 0.218 |
| delta | psi | mean | median | mad | rmse | sd |
|---|---|---|---|---|---|---|
| 0.1 | 0.2 | 0.569 | 0.626 | 0.282 | 0.379 | 0.379 |
| 0.1 | 0.5 | 0.345 | 0.461 | 0.440 | 0.588 | 0.532 |
| 0.1 | 0.8 | 0.164 | 0.269 | 0.579 | 0.739 | 0.600 |
| 0.5 | 0.2 | 0.618 | 0.602 | 0.108 | 0.143 | 0.143 |
| 0.5 | 0.5 | 0.600 | 0.610 | 0.127 | 0.159 | 0.160 |
| 0.5 | 0.8 | 0.647 | 0.639 | 0.210 | 0.258 | 0.255 |
| 0.7 | 0.2 | 0.679 | 0.630 | 0.148 | 0.205 | 0.190 |
| 0.7 | 0.5 | 0.633 | 0.621 | 0.116 | 0.155 | 0.153 |
| 0.7 | 0.8 | 0.636 | 0.619 | 0.136 | 0.175 | 0.172 |
Temporal misspecification
| delta | psi | mean | median | mad | rmse | sd |
|---|---|---|---|---|---|---|
| 0.1 | 0.2 | 0.040 | 0.079 | 0.381 | 0.475 | 0.474 |
| 0.1 | 0.5 | -0.018 | -0.036 | 0.484 | 0.571 | 0.562 |
| 0.1 | 0.8 | -0.044 | -0.083 | 0.521 | 0.604 | 0.590 |
| 0.5 | 0.2 | 0.097 | 0.076 | 0.152 | 0.232 | 0.234 |
| 0.5 | 0.5 | 0.072 | 0.076 | 0.152 | 0.194 | 0.193 |
| 0.5 | 0.8 | 0.081 | 0.068 | 0.236 | 0.315 | 0.316 |
| 0.7 | 0.2 | 0.142 | 0.095 | 0.170 | 0.283 | 0.281 |
| 0.7 | 0.5 | 0.086 | 0.088 | 0.137 | 0.202 | 0.202 |
| 0.7 | 0.8 | 0.085 | 0.081 | 0.158 | 0.221 | 0.222 |
| delta | psi | mean | median | mad | rmse | sd |
|---|---|---|---|---|---|---|
| 0.1 | 0.2 | 0.351 | 0.406 | 0.290 | 0.353 | 0.355 |
| 0.1 | 0.5 | 0.285 | 0.417 | 0.446 | 0.557 | 0.555 |
| 0.1 | 0.8 | 0.276 | 0.462 | 0.504 | 0.602 | 0.601 |
| 0.5 | 0.2 | 0.323 | 0.318 | 0.108 | 0.135 | 0.133 |
| 0.5 | 0.5 | 0.316 | 0.343 | 0.138 | 0.208 | 0.206 |
| 0.5 | 0.8 | 0.367 | 0.382 | 0.176 | 0.225 | 0.225 |
| 0.7 | 0.2 | 0.377 | 0.340 | 0.142 | 0.204 | 0.203 |
| 0.7 | 0.5 | 0.329 | 0.328 | 0.131 | 0.209 | 0.209 |
| 0.7 | 0.8 | 0.385 | 0.343 | 0.160 | 0.225 | 0.223 |
| delta | psi | mean | median | mad | rmse | sd |
|---|---|---|---|---|---|---|
| 0.1 | 0.2 | 0.627 | 0.631 | 0.237 | 0.283 | 0.284 |
| 0.1 | 0.5 | 0.421 | 0.479 | 0.375 | 0.499 | 0.469 |
| 0.1 | 0.8 | 0.250 | 0.325 | 0.504 | 0.640 | 0.538 |
| 0.5 | 0.2 | 0.635 | 0.631 | 0.109 | 0.142 | 0.138 |
| 0.5 | 0.5 | 0.619 | 0.621 | 0.122 | 0.151 | 0.151 |
| 0.5 | 0.8 | 0.674 | 0.683 | 0.193 | 0.231 | 0.220 |
| 0.7 | 0.2 | 0.693 | 0.654 | 0.144 | 0.191 | 0.168 |
| 0.7 | 0.5 | 0.638 | 0.621 | 0.110 | 0.146 | 0.141 |
| 0.7 | 0.8 | 0.664 | 0.659 | 0.142 | 0.182 | 0.171 |
Spatial misspecification
| delta | psi | mean | median | mad | rmse | sd |
|---|---|---|---|---|---|---|
| 0.1 | 0.2 | 0.061 | 0.048 | 0.205 | 0.250 | 0.248 |
| 0.1 | 0.5 | 0.077 | 0.074 | 0.167 | 0.208 | 0.208 |
| 0.1 | 0.8 | 0.038 | 0.048 | 0.164 | 0.209 | 0.201 |
| 0.5 | 0.2 | 0.105 | 0.108 | 0.131 | 0.163 | 0.164 |
| 0.5 | 0.5 | 0.103 | 0.096 | 0.122 | 0.152 | 0.152 |
| 0.5 | 0.8 | 0.082 | 0.094 | 0.126 | 0.157 | 0.157 |
| 0.7 | 0.2 | 0.109 | 0.127 | 0.132 | 0.166 | 0.166 |
| 0.7 | 0.5 | 0.113 | 0.124 | 0.116 | 0.146 | 0.146 |
| 0.7 | 0.8 | 0.100 | 0.077 | 0.133 | 0.163 | 0.164 |
| delta | psi | mean | median | mad | rmse | sd |
|---|---|---|---|---|---|---|
| 0.1 | 0.2 | 0.191 | 0.227 | 0.239 | 0.309 | 0.266 |
| 0.1 | 0.5 | 0.118 | 0.104 | 0.256 | 0.309 | 0.206 |
| 0.1 | 0.8 | 0.071 | 0.098 | 0.287 | 0.348 | 0.209 |
| 0.5 | 0.2 | 0.290 | 0.305 | 0.136 | 0.169 | 0.158 |
| 0.5 | 0.5 | 0.271 | 0.291 | 0.122 | 0.163 | 0.144 |
| 0.5 | 0.8 | 0.251 | 0.250 | 0.148 | 0.186 | 0.158 |
| 0.7 | 0.2 | 0.313 | 0.332 | 0.103 | 0.137 | 0.132 |
| 0.7 | 0.5 | 0.305 | 0.329 | 0.099 | 0.132 | 0.125 |
| 0.7 | 0.8 | 0.297 | 0.301 | 0.109 | 0.141 | 0.131 |
| delta | psi | mean | median | mad | rmse | sd |
|---|---|---|---|---|---|---|
| 0.1 | 0.2 | 0.274 | 0.314 | 0.338 | 0.420 | 0.265 |
| 0.1 | 0.5 | 0.160 | 0.168 | 0.442 | 0.474 | 0.177 |
| 0.1 | 0.8 | 0.104 | 0.086 | 0.497 | 0.532 | 0.192 |
| 0.5 | 0.2 | 0.503 | 0.517 | 0.131 | 0.175 | 0.146 |
| 0.5 | 0.5 | 0.423 | 0.433 | 0.183 | 0.224 | 0.138 |
| 0.5 | 0.8 | 0.371 | 0.390 | 0.232 | 0.269 | 0.143 |
| 0.7 | 0.2 | 0.543 | 0.555 | 0.098 | 0.128 | 0.115 |
| 0.7 | 0.5 | 0.496 | 0.512 | 0.115 | 0.151 | 0.111 |
| 0.7 | 0.8 | 0.467 | 0.470 | 0.143 | 0.178 | 0.119 |